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Optimal Sequential Flows

Hugo Gimbert, Corto Mascle, Patrick Totzke

TL;DR

The paper addresses maximal sequential flow in time-varying networks when capacity labels are drawn from a finite set, allowing unbounded optimal values. It introduces a flow semigroup built from max-min abstractions and iterates capacity words via a $\sharp$ operator, enabling a two-stage PSPACE approach: detect unboundedness via unboundedness witnesses, and, if bounded, compute the finite maximum using $\sharp$-expressions with bounded height. Central to the method are summaries and $\sharp$-summaries, which provide compact representations and polynomial-space bounds through a general factorization theorem for finite semigroups and its application to the flow setting. The framework extends to fairness over multiple edges and to regular constraints, yielding PSPACE algorithms with explicit exponential bounds that depend on the graph size and automaton parameters, thereby offering scalable, algebraic techniques for dynamic network flows.

Abstract

We provide a new algebraic technique to solve the sequential flow problem in polynomial space. The task is to maximize the flow through a graph where edge capacities can be changed over time by choosing a sequence of capacity labelings from a given finite set. Our method is based on a novel factorization theorem for finite semigroups that, applied to a suitable flow semigroup, allows to derive small witnesses. This generalizes to multiple in/output vertices, as well as regular constraints.

Optimal Sequential Flows

TL;DR

The paper addresses maximal sequential flow in time-varying networks when capacity labels are drawn from a finite set, allowing unbounded optimal values. It introduces a flow semigroup built from max-min abstractions and iterates capacity words via a operator, enabling a two-stage PSPACE approach: detect unboundedness via unboundedness witnesses, and, if bounded, compute the finite maximum using -expressions with bounded height. Central to the method are summaries and -summaries, which provide compact representations and polynomial-space bounds through a general factorization theorem for finite semigroups and its application to the flow setting. The framework extends to fairness over multiple edges and to regular constraints, yielding PSPACE algorithms with explicit exponential bounds that depend on the graph size and automaton parameters, thereby offering scalable, algebraic techniques for dynamic network flows.

Abstract

We provide a new algebraic technique to solve the sequential flow problem in polynomial space. The task is to maximize the flow through a graph where edge capacities can be changed over time by choosing a sequence of capacity labelings from a given finite set. Our method is based on a novel factorization theorem for finite semigroups that, applied to a suitable flow semigroup, allows to derive small witnesses. This generalizes to multiple in/output vertices, as well as regular constraints.

Paper Structure

This paper contains 19 sections, 35 theorems, 22 equations, 5 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Sequential Flows
  3. Solving the
  4. Diving into the flow semigroup
  5. A general factorization theorem for finite semigroups
  6. Application to the flow semigroup and iterations
  7. Extensions
  8. Fair flows along multiple edges
  9. Regular constraints
  10. Proofs of Section \ref{['sec:seqflow']}
  11. Proof of Lemma \ref{['lem:idempotent']}
  12. Proof of Lemma \ref{['lem:sharpsharp']}
  13. Proof of Lemma \ref{['lem:sufficientwitness']}
  14. Proof of Lemma \ref{['lem:algoquali']}
  15. Checking whether $x_s \to_\ell x_t$ in polynomial space
  16. ...and 4 more sections

Key Result

Lemma 5

Let $e\in \mmsm^{V\times V}$ such that $e=e^2$, and $v,v' \in V$ such that $e(v,v')>0$. For $n\geq 1$, let $K_n$ denote the optimal flow value from $v$ to $v'$ in the pipeline $e^n$. Exactly one of the following holds.

Figures (5)

  • Figure 1: The two capacities $a,b$ from \ref{['ex:intro']}, the pipeline, and the optimal flow for $abba$.
  • Figure 2: A flow of value $n+1$ through the pipeline $ab^{n+1}a$.
  • Figure 3: Capacities $c,d,e$ from \ref{['ex:figurec']}, pipeline $cccc$ and its maximal flow, and pipeline $ec$.
  • Figure 4: The pipelines from Example \ref{['ex:nested']}. The pipeline for $ab^nc$ and its shortened representation (seen on the left) is iterated another $n$ times in the pipeline for $(ab^nc)^na$ (seen on the right).
  • Figure 5: The figure illustrates Lemma \ref{['lem:idempotent']}, which classifies the three possible long-term behaviours of an edge $(v,v')$ of an idempotent $e$, when $e(v,v')\neq 0$. Case i) is $e(v,v') =\omega$ and case ii) and iii) occur when $e(v,v') =1$.

Theorems & Definitions (56)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Lemma 5: Flow-carrying Edges of idempotent elements
  • Definition 6: iteration of an idempotent
  • Lemma 7
  • Definition 8: flow semigroup
  • Example 9
  • Definition 10: Unboundedness witness
  • ...and 46 more